Just multiply it out in Fourier, where partial=ik:
bigl[(−k2+m2)gmunu+kmuknubigr]frac−gnulambda+m−2knuklambdak2−m2=frac1k2−m2bigl(−(−k2+m2)deltamulambda+(−k2+m2)m−2kmuklambda−kmuklambda+kmuk2klambdam−2bigr)=deltamulambda
which is a function of k. But converting back to position space, 1(k)=delta(x).
This proves that D is a solution. To be the solution, you usually have to impose boundary conditions, etc. In this case, there are no solutions to bigl[(−k2+m2)gmunu+kmuknubigr]fnu=0: the corresponding equation in Fourier is (−k2+m2)gmunu+kmuknu=0, and contracting with gmunu gives 0=d(−k2+m2)+k2=dm2−(d−1)k2, where d is the dimension of spacetime, so k2=fracdd−1m2, but −frac1d−1m2gmunu+kmuknu cannot equal 0, as kmuknu cannot be an invertible matrix. So D is the only solution.
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